Distributed Algorithms for Minimum Degree Spanning Trees

TL;DR

This paper introduces the first efficient distributed algorithms for the minimum degree spanning tree problem, achieving near-optimal approximation ratios with manageable round complexities in a distributed setting.

Contribution

The paper presents two novel distributed approximation algorithms for MDST, improving the approximation factor and round complexity over prior sequential solutions.

Findings

First algorithm achieves O(d log n) maximum degree with O((D + √n) log^2 n) rounds.

Second algorithm improves approximation to O(d + log n) with additional polylogarithmic factors.

These are the first efficient distributed algorithms for the MDST problem.

Abstract

The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $\hat d = O(d\log{n})$. It requires $O((D + \sqrt{n}) \log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexity for the related minimum spanning tree problem. Our second result refines this approximation factor by constructing a tree with maximum degree $\hat d = O(d + \log{n})$, though at the cost of additional polylogarithmic factors in the round complexity. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990's, our results are first efficient distributed solutions for this problem.